# reference help needed on a fact about decomposition of representations

For simplicity, let $G_n$ be $GL_n(\mathbb{R})$, $\mathfrak{g}_n$ be its Lie algebra. $K_n$ be $O(n)$. I want to know any reference about the following statement.

For any irreducible admissible $(\mathfrak{g}_n\oplus \mathfrak{g}_m,K_n\times K_m)$ module $V$, there exist an irreducible admissible $(\mathfrak{g}_n, K_n)$ module $U$, and an irreducible admissible $(\mathfrak{g}_m, K_m)$ module $W$, such that $V$ is isomorphic to $U\otimes W$. Both $U$ and $W$ are uniquely determined by $V$ up to isomorphism.

Similar of this is true for representations of finite groups, smooth representations of p-adic groups. It seems to me that this is also true for real reductive groups, and I can't find it in any standard reference. I appreciate a lot if anyone would provide some related paper or book.

• For the analogous statement regarding unitary irreducible representations, see Dixmier's book on $C^*$-algebras: combine Proposition 13.1.8 (every unitary irreducible rep of a product of two type I groups, decomposes as a tensor product of irreducible rep's) with Theorem 15.5.6 ($GL_n(\mathbb{R})$ is type I). – Alain Valette Sep 2 '11 at 6:40

See Theorem 1.1 and its proof. There the authors show that each irreducible Harish-Chandra $(\mathfrak{g}_1 \times \mathfrak{g}_2, K_1 \times K_2)$-module has a unique decomposition into a tensor product of irreducible Harish-Chandra $(\mathfrak{g}_i, K_i)$-modules. (Here by a Harish-Chandra $(\mathfrak{g},K)$-module we mean a $(\mathfrak{g},K)$-module that is admissible and finitely generated as a $U(\mathfrak{g}_\mathbb{C})$-module.)